Suppose we have the random walk with drift model
$$x_t = \delta+x_{t-1}+w_t$$ for $t = 1, 2, ...$ with initial condition $x_0=0$ and where $w_t$ is white noise. The constant $\delta$ is called the drift, when $\delta$=0, the equation above is simply a random walk. $t$ is the time. We may rewrite this as the cumulative sum of white noise variates:
$$x_t = \delta t+\sum_{j=1}^tw_j$$
What I have tried to show this:
$$\delta t+\sum_{j=1}^tw_j=\delta+x_{t-1}+w_t \\ x_{t-1}=\delta(t-1)+\sum_{j=1}^tw_j-w_t$$
If I remember summation tricks properly, then
$$\sum_{j=1}^t w_j-w_t = \sum_{j=1}^{t+1}w_j$$
So we have
$$x_{t-1} = \delta(t-1)+\sum_{j=1}^{t+1}w_j$$